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Midterm I Solutions

Midterm I Solutions - MATH 74 MIDTERM 1 1(a False(No matter...

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MATH 74 MIDTERM 1 1(a). False. (No matter what * is, and no matter what a S is, the equation a * a = a * a is always true, for example.) 1(b). True. For any integer k , the real number sin( πk 2 ) is an integer: it is either 0, 1, or - 1 depending on whether k is even, one more than a multiple of four, or one less than a multiple of four. So the formula for a * b actually does give an integer result for any integers a and b . 1(c). False. The problem is that any given rational number has many different representa- tions as a quotient a b , and the result of the formula depends on which one you pick. (I did not specify whether a, b, c, d are integers, or are themselves rational numbers; either way, this is a problem.) For example, suppose I want to compute 1 2 * 1. If write 1 as 1 1 the formula says 1 2 * 1 is 1 2 . If I write 1 as 13 13 the formula says 1 2 * 1 is 1 26 . Because of this ambiguity, the given formula does not define a binary operation on Q .

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